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1 VECTOR MOMENTS

2 Question 1 (**) The vectors i, j and k are unit vectors mutually perpendicular to one another. Relative to a fixed origin O, a light rigid rod has its ends located at the points 0, 7,4 B 4,9, 2. A force F acts at the midpoint of the rod. A( ) and ( ) In standard vector notation, when = ( + k ) F 4i j k N, where k is a constant, the magnitude of the moment of F about B has magnitude 9 17 Nm. Show clearly that k = ± 1 proof

3 Question 2 (**) The unit vectors i, j and k are oriented in the positive x direction, positive y direction and positive z direction, respectively. Three forces ( + + ) 3 4 N a N A 1, 2, a 6,2,8 i j k, ( + ) i j k and ( a + a ) constant, act at the points ( ), B ( ) and ( 1,0, 1) i 2 j k N, where a is a C, respectively. Distances are measured in m, relative to a fixed origin O. a) Given that the moment of the system of the three forces about C is zero, determine the value of a. b) Find the magnitude of the moment of the system of the three forces about O, showing clearly that its value is independent of a. M456-F, a = 15, G 0 = 2 2 Nm

4 Question 3 (**) The vectors i, j and k are unit vectors mutually perpendicular to one another. Three forces, F 1, F 2 and F 3, act on a rigid body at the points with position vectors r 1, r 2 and r 3, respectively. This information is summarised below. F = ( i + j k ) N acting at ( ) r 1 = i + j k m F = ( i j k ) N acting at = ( + + ) 2 2 r i j k m 2 2 F = ( i j+ k) N acting at = ( + ) r i j k m 1 3 Show that the system is equivalent to a couple and find the magnitude of the vector moment of this couple. G = Nm

5 Question 4 (**) The standard unit vectors i and j are oriented in the positive x direction and positive y direction, respectively. Three forces F = ( i + bj ), F = ( ai + bj ) and = ( b + ) 1 4 N N F3 10 i 3j N, where a and b are scalar constants, are acting at the points A ( 1,2 ), ( 4, 2) C ( 3, 5), respectively. B and a) Given that the resultant of the three forces is zero, determine the magnitude and direction of the total moment of these three forces about O. b) Find, by direct calculation, the magnitude and direction of the total moment of these three forces about C. M456-A, G 64 Nm, clockwise, G 64 Nm, clockwise O = C =

6 Question 5 (**+) The vectors i, j and k are unit vectors mutually perpendicular to one another. Three forces, F 1, F 2 and F 3, act on a rigid body at the points with position vectors r 1, r 2 and r 3, respectively. This information is summarised below. F = ( i + j k) N acting at ( ) r 1 = i + j+ k m F = ( i j k ) N acting at = ( ) r i k m 2 4 F = ( i + j+ k) N acting at = ( + ) r1 2i j 5k m The system of the three forces is equivalent to a single force R acting at the point 2i k m, together with a couple of moment G. with position vector ( ) Determine R and G in vector form. R = 3i + 5j 2k, G = 15i + 10j k

7 Question 6 (***) The vectors i, j and k are unit vectors mutually perpendicular to one another. Three forces F = ( i j+ k ), = ( ) The force N F2 i 2k N and F 3 act on a rigid body. F acts through the point with position vector ( ) i + j+ 2k m, the force F 2 acts through the point with position vector ( i j )m and the force F 3 acts through the point with position vector ( + + ) 2i j k m. The system of the three forces reduce to a couple G. a) Determine G. The line of action of F 3 is changed so that the system of the three forces now reduces to the couple ( + ) 4i j k Nm. b) Find a vector equation of the new line of action of F 3. ( 5i + j + 3k ) Nm, r = ( 8 i j ) + λ ( 4i + j+ k )

8 Question 7 (***) The vectors i, j and k are unit vectors mutually perpendicular to one another. Three forces F = ( i j+ k ), = ( + ) N F2 2i 4j 5k N and F 3 act on a rigid body. The force F 1 acts through the point with position vector ( + ) F acts through the point with position vector ( 3i 3j+ 5k ) m. 2 2j 4k m and the force The system of the three forces is in equilibrium. a) Find a vector equation of the line of action of F 3. The force F 3 is replaced by a force F 4 acting through the point ( i j )m. The system of F 1, F 2 and F 4 is now equivalent to a single force ( i j k ) N acting through the point ( + + ) 2i j k m, together with a couple G. b) Determine the magnitude of G. ( ) + λ ( 3 ) r = i + j i j+ k, G = Nm

9 Question 8 (***) The unit vectors i, j and k are oriented in the positive x direction, positive y direction and positive z direction, respectively. Three forces F = ( i j ), F = ( i j+ k ) and = ( ) N N are acting at the points A1 ( 1,1,0 ), A 2 ( 2,0,5 ) and 3 ( 6,2,1 ) a) Show that the system reduces to a single force F. b) Find an equation of the line of action of F. F2 3j 4k N, A, respectively. M456-B, r = 4i + 3j+ λ ( 7i 2k )

10 Question 9 (***) The standard unit vectors i and j are oriented in the positive x direction and positive y direction, respectively. Three forces F1 = 4i + bj, F2 = 3ai + 2bj and F3 = 10bi + 3j, where a and b are scalar constants, are acting at the points A 1( 1,2 ), A 2 ( 4, 2) and 3 ( 3, 5 ) A, respectively. a) Determine the magnitude and direction of the total moment of these three forces about O. b) Find, by direct calculation, the magnitude and direction of the total moment of these three forces about C. M456-A, G 64 Nm, clockwise, G 64 Nm, clockwise O = C =

11 Question 10 (***+) The standard unit vectors i and j are oriented in the positive x direction and positive y direction, respectively. Three forces ( 3a + 1) i + 3j N, ( a 10) i 2j N and + ( a) i 1 j N, where a is a constant, act at the points A ( 1,2 ), B ( 2,0) and ( 4, 1) C, respectively. Distances are measured in m, relative to a fixed origin O. a) Given that the system of the three forces reduces to a couple about O, find the magnitude and direction of this couple. b) Given instead that the system of the three forces reduces to single force F, determine the equation of the line of action of F. M456-C, G 0 = 18 Nm, anticlockwise, y = 1 4 x

12 Question 11 (***+) The vectors i, j and k are unit vectors mutually perpendicular to one another. Three forces F = ( i + j+ k ), = ( + ) The force N F2 i 2k N and F 3 act on a rigid body. F acts through the point with position vector ( + + ) F acts through the point with position vector ( i j+ k )m. i 2j 2k m and the force The system of the three forces is in equilibrium. Show that the line of action of F 3 passes through the point with position vector 2k. proof

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